In practice, the \(X_i\)  are shortrun MCMC samples from an EBM and the \(Z_i\)  are unobserved. The full formulation of cooperative learning involves latent MCMC using the conditional distribution \(p(Z| X_i ;\phi )\)  to infer each \(Z_i\) . In practice, since the \(X_i\)  are shortrun samples that are paired with \(Z_i\)  through the process of rejuvenation, one can roughly approximate the latent MCMC process with an identity function and obtain a gradient (REF ). We find that this works as well or better than including a latent sampling step on \(Z\) , and it is consistent with the original implementation from [1]}.
